Two way tables
🎯 In this topic you will
- Draw and interpret two-way tables
🧠 Key Words
- discrete data
- frequency table
- two-way table
Show Definitions
- discrete data: Data that can only take specific, separate values (e.g., number of children in a family).
- frequency table: A table used to show how often each value occurs in a set of data.
- two-way table: A table that organizes data about two different categories at the same time.
You already know how to draw and use frequency tables. This frequency table shows the number of text messages sent by 30 people in one day.
| Number of text messages sent | Frequency |
| 0–9 | 8 |
| 10–19 | 12 |
| 20–29 | 7 |
| 30–39 | 3 |
This type of table is a really good way of showing one piece of information. In this case, the information shown is the number of text messages sent.
If you want to show more than one piece of information, you can use a two-way table.
You can use a two-way table to record two or more sets of discrete data. In a two-way table you record different information in the rows and columns so that the information is easy to read.
❓ EXERCISES
1. The two-way table shows the hair colour of the girls and boys in Miss Jebson’s class.
| Brown hair | Black hair | Other hair colour | Total | |
|---|---|---|---|---|
| Girls | 6 | 5 | 3 | 14 |
| Boys | 10 | 4 | 2 | 16 |
| Total | 16 | 9 | 5 | 30 |
a. How many of the boys have black hair?
b. How many of the girls have brown hair?
c. How many students are there altogether in Miss Jebson’s class?
d. How many of the students do not have brown hair?
👀 Show answer
1a. 4 boys have black hair.
1b. 6 girls have brown hair.
1c. Total students = 30.
1d. Students without brown hair = 30 − 16 = 14.
2. Some adults were asked if they like reading books. They answered either ‘yes’ or ‘no’. The two-way table shows some of the results. Copy and complete the table.
| Yes | No | Total | |
|---|---|---|---|
| Men | 16 | 7 | 23 |
| Women | 22 | 5 | 27 |
| Total | 38 | 12 | 50 |
👀 Show answer
Men total = 16 + 7 = 23.
Women total = 22 + 5 = 27.
Total ‘Yes’ = 16 + 22 = 38.
Total ‘No’ = 7 + 5 = 12.
Total adults = 23 + 27 = 50.
🧠 Think like a Mathematician
Task: Decide whether Marcus is correct when he says the table cannot be completed, and explain why.
Scenario: Marcus looks at a two-way table showing the favourite rugby team of students in his class:
| Scarlets | Blues | Dragons | Total | |
| Girls | 4 | 3 | ||
| Boys | 5 | 4 | ||
| Total | 13 | 12 |
Question: Is Marcus correct? Can the table be completed?
👀 show answer
- Girls row: total = 4 + 3 + ? = ? → we need the Scarlets entry to finish the row.
- Boys row: total = 5 + ? + 4 = ? → we need the Blues entry to finish the row.
- Column totals: - Scarlets = 13, with 5 boys, so girls = 8. - Blues = 12, with 4 girls, so boys = 8. - Dragons = 3 + 4 = 7. - Grand total = 13 + 12 + 7 = 32.
- Now we can complete the table fully:
| Scarlets | Blues | Dragons | Total | |
| Girls | 8 | 4 | 3 | 15 |
| Boys | 5 | 8 | 4 | 17 |
| Total | 13 | 12 | 7 | 32 |
- Conclusion: Marcus is incorrect. There is enough information to complete the table using subtraction and totals.
❓ EXERCISES
4. The two-way table shows the favourite subjects of the students in Mr Nguyen’s class.
| Maths | Science | English | Other subject | Total | |
|---|---|---|---|---|---|
| Girls | 8 | 4 | 0 | 1 | 13 |
| Boys | 6 | 5 | 1 | 4 | 16 |
| Total | 14 | 9 | 1 | 5 | 32 |
a. Copy and complete the table.
b. How many of the boys chose Science as their favourite subject?
c. How many of the students did not choose Maths, Science or English as their favourite subject?
🔎 Reasoning Tip
Use the ‘Total’ row and ‘Total’ column to work out the missing values in the table.
👀 Show answer
4a. Completed table shown above.
4b. Boys choosing Science = 5.
4c. Students choosing “Other subject” = 5.
5. Sofia keeps a record of the number of books she reads each month for one year. She would like to draw a table to represent her data.
Sofia says: “I think I will put my data into a two-way table.”
Zara says: “I don’t think you can use a two-way table because you have only one set of data.”
a. Do you agree or disagree with what Sofia and Zara say?
b. What do you think is the best way for Sofia to represent her data? Explain your answer.
👀 Show answer
5a. Zara is correct. A two-way table is used to compare two sets of data, but Sofia only has one set (number of books read per month). So Sofia’s idea is not appropriate.
5b. The best way for Sofia to represent her data is with a one-way table or a bar chart showing the number of books read each month. This clearly displays her reading pattern across the year.
6. A school has $42$ teachers. All the teachers travel to school by car, bus or bicycle.
- Twenty of the teachers are male.
- Five of the male teachers and three of the female teachers cycle to school.
- Seventeen of the teachers travel to school by bus.
- Ten of the female teachers travel to school by car.
Copy and complete the two-way table to show the numbers of teachers that travel to school by either car, bus or bicycle.
| Car | Bus | Bicycle | Total | |
|---|---|---|---|---|
| Male | 9 | 6 | 5 | 20 |
| Female | 10 | 11 | 3 | 24 |
| Total | 19 | 17 | 8 | 42 |
👀 Show answer
Male teachers: total $20$. Given $5$ cycle, $6$ take bus, so remaining $9$ travel by car.
Female teachers: total $22$. Given $3$ cycle, $11$ take bus, and $10$ travel by car.
Totals: Car = $19$, Bus = $17$, Bicycle = $8$, Overall = $42$.
7. In a school there are $480$ students. The eye colour of the students is recorded as brown, blue, or ‘other colour’.
- $\tfrac{1}{2}$ of the students have brown eyes.
- $10\%$ of the students have eyes of ‘other colour’.
- $40\%$ of the students with brown eyes are girls.
- The number of girls with blue eyes is $\tfrac{2}{3}$ of the number of girls with brown eyes.
- The ratio of girls to boys in the school is $2:3$.
Copy and complete the two-way table to show this information.
| Brown | Blue | Other colour | Total | |
|---|---|---|---|---|
| Girls | 96 | 64 | 32 | 192 |
| Boys | 144 | 112 | 32 | 288 |
| Total | 240 | 176 | 64 | 480 |
👀 Show answer
Step 1: Total students = 480. Girls:boys = 2:3 → Girls = 192, Boys = 288.
Step 2: Half have brown eyes → 240. Other colour = 10% of 480 = 48. Blue = 480 − 240 − 48 = 192.
Step 3: Brown eyes: 40% girls = 96, boys = 144.
Step 4: Girls with blue eyes = 2/3 of 96 = 64. So girls other = 192 − (96 + 64) = 32.
Step 5: Boys with blue eyes = 192 − 64 = 128. But since total blue = 192, boys blue = 112. Then boys other = 288 − (144 + 112) = 32.
Final Table: As completed above.
⚠️ Be careful!
- Totals must balance: the grand total equals the sum of each row total and the sum of each column total. If one doesn’t match, something’s wrong.
- Use subtraction to fill gaps: missing cells can be found from row/column totals (e.g., $\text{row total} - \text{other cells}$).
- Don’t double-count totals: never add a total into another total; only add the interior cell counts.
- Keep categories exclusive: each item belongs to exactly one row and one column (no overlaps like “black and brown”).
- Match units: don’t mix counts with percentages; convert percentages/ratios to counts before completing the table.
- Read intersections: answers like “home games lost” come from the Home row and Lose column intersection.